TSM - Assignment.

Collect the data for real and nominal yields from the Federal Reserve covering the period Jan 2005-August 2022.

import pandas as pd
from helpers import *
import plotly.graph_objects as go
import plotly.express as px
from statsmodels.tsa.stattools import adfuller
import chart_studio.plotly as py
import plotly.io as pio
import plotly.offline as pyo
import numpy as np

#
pio.renderers.default = "notebook"
# collect data from files
nominal_yields = pd.read_excel('nominal_yields.xlsx').dropna()
real_yields = pd.read_excel('real_yields.xlsx').dropna()

# show the first 5 rows of nominal yields
nominal_yields.head()

	Date	BETA0	BETA1	BETA2	BETA3	SVEN1F01	SVEN1F04	SVEN1F09	SVENF01	SVENF02	...	SVENY23	SVENY24	SVENY25	SVENY26	SVENY27	SVENY28	SVENY29	SVENY30	TAU1	TAU2
0	2005-01-03	6.189826e-06	2.277828	2.659254	16.057365	3.4022	4.3567	5.7028	3.1680	3.5462	...	5.0962	5.0874	5.0729	5.0535	5.0295	5.0014	4.9697	4.9348	1.436363	13.431858
1	2005-01-04	2.549831e-09	2.259019	3.046883	16.223499	3.5231	4.4071	5.7558	3.2872	3.6552	...	5.1584	5.1495	5.1350	5.1155	5.0913	5.0631	5.0312	4.9960	1.393715	13.471621
2	2005-01-05	4.330379e-05	2.259806	3.089485	16.079062	3.5391	4.3983	5.7099	3.3009	3.6710	...	5.1227	5.1131	5.0981	5.0780	5.0535	5.0249	4.9928	4.9574	1.412061	13.445450
3	2005-01-06	1.834049e-12	2.257302	2.977634	16.110743	3.5090	4.4031	5.7234	3.2678	3.6475	...	5.1278	5.1183	5.1032	5.0831	5.0586	5.0299	4.9977	4.9623	1.423321	13.430211
4	2005-01-07	1.780984e-12	2.304075	2.956248	16.007722	3.5372	4.4163	5.6898	3.2920	3.6785	...	5.1104	5.1006	5.0854	5.0651	5.0405	5.0118	4.9796	4.9442	1.476347	13.455895

5 rows × 100 columns

# show the first 5 rows of real yields
real_yields.head()

	Date	BETA0	BETA1	BETA2	BETA3	BKEVEN02	BKEVEN03	BKEVEN04	BKEVEN05	BKEVEN06	...	TIPSY11	TIPSY12	TIPSY13	TIPSY14	TIPSY15	TIPSY16	TIPSY17	TIPSY18	TIPSY19	TIPSY20
0	2005-01-03	-1.504322	1.565163	-3167.402774	3176.711476	2.5935	2.5925	2.5779	2.5709	2.5767	...	1.8002	1.8722	1.9313	1.9784	2.0144	2.0401	2.0565	2.0643	2.0644	2.0575
1	2005-01-04	-0.973675	1.184436	-3682.065233	3690.256864	2.5802	2.5909	2.5779	2.5701	2.5747	...	1.8629	1.9333	1.9910	2.0369	2.0718	2.0967	2.1123	2.1197	2.1196	2.1127
2	2005-01-05	-0.692399	0.976122	-3186.398677	3193.908160	2.5502	2.5701	2.5611	2.5536	2.5569	...	1.8764	1.9459	2.0028	2.0479	2.0821	2.1062	2.1213	2.1281	2.1276	2.1205
3	2005-01-06	-0.344569	0.609703	-1901.199337	1907.956756	2.5499	2.5726	2.5662	2.5605	2.5645	...	1.8695	1.9399	1.9974	2.0428	2.0772	2.1015	2.1167	2.1237	2.1234	2.1168
4	2005-01-07	-0.535426	0.854999	-2554.392640	2561.606742	2.5209	2.5466	2.5423	2.5372	2.5405	...	1.9012	1.9703	2.0267	2.0715	2.1053	2.1292	2.1441	2.1509	2.1504	2.1435

5 rows × 127 columns

# select specific maturities
nominal_yields_2_10y_eom = selectYieldsMaturities(nominal_yields,type='nominal',FD=False)
real_yields_2_10y_eom = selectYieldsMaturities(real_yields,type='real',FD=False)

# plot nominal data
plotYield(nominal_yields_2_10y_eom,columns=['SVENY02','SVENY03','SVENY05','SVENY07','SVENY10'],type='Nominal',FD=False)

# plot real yield data (TIPS)
plotYield(real_yields_2_10y_eom,columns=['TIPSY02','TIPSY03','TIPSY05','TIPSY07','TIPSY10'],type='Real',FD=False)

We have extracted and plotted the data, but before proceeding with PCA, it is usually well-practice to ensure stationarity of the data under consideration. From the figures of nominal and real yields respectively, some clear trends which for the statistical inference of data analysis should be removed (or limited as much as possible) for a correct interpretation of the results.

# check for stationarity in nominal yields
ADFtest(nominal_yields_2_10y_eom,type='nominal',maturity='2y')

ADF Statistic: -1.784198
p-value: 0.388335
Critical Values:
	1%: -3.476
	5%: -2.881
	10%: -2.577
Non-Stationary timeserie

(-1.7841977737615378, 0.3883348001847712)

# let's try for 5y yields
ADFtest(nominal_yields_2_10y_eom,type='nominal',maturity='5y')

ADF Statistic: -1.843918
p-value: 0.358886
Critical Values:
	1%: -3.475
	5%: -2.881
	10%: -2.577
Non-Stationary timeserie

(-1.8439183137699224, 0.3588855951201053)

# let's try for 10y yields
ADFtest(nominal_yields_2_10y_eom,type='nominal',maturity='10y')

ADF Statistic: -1.902721
p-value: 0.330782
Critical Values:
	1%: -3.475
	5%: -2.881
	10%: -2.577
Non-Stationary timeserie

(-1.9027209249148123, 0.3307821864201821)

# nominal yields are non-stationary, so we re-take the original data, apply first difference and then take only the month end
nominal_yields_2_10y_eom_shifted = selectYieldsMaturities(nominal_yields,type='nominal',FD=True)

# check stationarity after first difference (only for the 2y)
ADFtest(nominal_yields_2_10y_eom_shifted,type='nominal',maturity='2y')

ADF Statistic: -7.123238
p-value: 0.000000
Critical Values:
	1%: -3.477
	5%: -2.882
	10%: -2.578
Stationary timeserie

(-7.123237767881004, 3.6760004763770164e-10)

# let's plot the nominal yields stationary data
plotYield(nominal_yields_2_10y_eom_shifted,columns=['SVENY02','SVENY03','SVENY05','SVENY07','SVENY10'],type='Nominal',FD=True)

# check for stationarity in real yields
ADFtest(real_yields_2_10y_eom,type='real',maturity='2y')

ADF Statistic: -2.347080
p-value: 0.157233
Critical Values:
	1%: -3.476
	5%: -2.881
	10%: -2.577
Non-Stationary timeserie

(-2.347080382879149, 0.15723267907692623)

# let's try with 5y
ADFtest(real_yields_2_10y_eom,type='real',maturity='5y')

ADF Statistic: -2.058364
p-value: 0.261583
Critical Values:
	1%: -3.475
	5%: -2.881
	10%: -2.577
Non-Stationary timeserie

(-2.05836398030184, 0.2615826889940359)

# let's try with 10y
ADFtest(real_yields_2_10y_eom,type='real',maturity='10y')

ADF Statistic: -1.859705
p-value: 0.351248
Critical Values:
	1%: -3.476
	5%: -2.882
	10%: -2.577
Non-Stationary timeserie

(-1.8597053177414344, 0.351247809531534)

# real yields are non-stationary, so we re-take the original data, apply first difference and then take only the month end
real_yields_2_10y_eom_shifted = selectYieldsMaturities(real_yields,type='real',FD=True)

# check stationarity after first difference (only for the 2y)
ADFtest(real_yields_2_10y_eom_shifted,type='real',maturity='2y')

ADF Statistic: -13.979960
p-value: 0.000000
Critical Values:
	1%: -3.475
	5%: -2.881
	10%: -2.577
Stationary timeserie

(-13.97996046052464, 4.194864820791476e-26)

# let's plot the nominal yields stationary data
plotYield(real_yields_2_10y_eom_shifted,columns=['TIPSY02','TIPSY03','TIPSY05','TIPSY07','TIPSY10'],type='Real',FD=True)

Now we have done all the preliminary analysis entailing first differencing the data to ensure stationarity. At this point, we can start with the principal components analysis (PCA) using the stationary data.

# PCA